Maximizing directed cycles in tournaments

Abstract

Determining the combinatorial structures that maximize the number of prescribed substructures is a central theme in extremal combinatorics. Grzesik, Kr\'al', Lov\'asz and Volec showed that when $\ell$ is not divisible by $4$, the random tournament contains asymptotically the most directed cycles of length $\ell$ among all $n$-vertex tournaments. In the paper, we resolve the remaining cases where $\ell$ is divisible by $4$. We show that, in this regime, the so-called carousel tournament asymptotically maximizes the number of directed $\ell$-cycles among all $n$-vertex tournaments, and in particular contains strictly more such cycles than the random tournament. This confirms the conjecture of Bartley and Day.